Edge-Minimum Saturated k-Planar Drawings

Abstract

For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D\in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D^{\prime }\notin \mathcal{D}$ - this is analogous to saturated graphs in a graph class as introduced by Turán (1941) and Erdős, Hajnal, and Moon (1964). We focus on $k$-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most $k$ times, and the classes $\mathcal{D}$ of all $k$-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated $k$-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all $n$-vertex saturated $k$-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest $n$-vertex saturated $k$-planar drawings have $\frac{2}{k-(k\mathrm{mod}2)}(n-1)$ edges for any $k\ge 4$, while if all that is forbidden, the sparsest such drawings have $\frac{2(k+1)}{k(k-1)}(n-1)$ edges for any $k\ge 6$.

Appears in the Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021). This version merges the previous version with some parts of arXiv:2012.02281